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\begin{equation}
\begin{aligned}
&amp;2 a \sum_{n=0}^{\infty} (-1)^n \frac{(2n+1)  x^{2n+1}}{(2n+1)!}+\sum_{n=0}^{\infty} (n^2-n) d_n x^n+\sum_{k=2}^{\infty} d_{k-2} x^k=0\\
&amp;\to 2 a \sum_{n=0}^{\infty} (-1)^n \frac{(2n+1)  x^{2n+1}}{(2n+1)!}+\sum_{n=2}^{\infty} [n(n-1)d_n+d_{n-2}] x^n=0.
\end{aligned}\tag{5.5.12}
\end{equation}
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